A density-equalizing map (DEM) serves as a powerful technique for creating shape deformations with the area changes reflecting an underlying density function. In recent decades, DEMs have found widespread applications in fields such as data visualization, geometry processing, and medical imaging. Traditional approaches to the DEM primarily rely on iterative numerical solvers for diffusion equations or optimization-based methods that minimize handcrafted energy functionals. However, these conventional techniques often face several challenges: they may suffer from limited accuracy, produce overlapping artifacts in extreme cases, and require substantial algorithmic redesign when extended from 2D to 3D, due to the derivative-dependent nature of their energy formulations. In this work, we proposed a novel learning-based density-equalizing mapping framework (LDEM) using deep neural networks. Specifically, we introduced a loss function that enforces density uniformity and geometric regularity, and utilized a hierarchical approach to predict the transformations at both the coarse and dense levels. Our method demonstrated superior density-equalizing and bijectivity properties compared to prior methods for a wide range of simple and complex density distributions, and can be easily applied to surface remeshing with different effects. Also, it generalizes seamlessly from 2D to 3D domains without structural changes to the model architecture or loss formulation. Altogether, our work opens up new possibilities for scalable and robust computation of density-equalizing maps for practical applications.
Citation: Yanwen Huang, Lok Ming Lui, Gary P. T. Choi. Learning-based density-equalizing map[J]. AIMS Mathematics, 2025, 10(11): 25756-25790. doi: 10.3934/math.20251140
A density-equalizing map (DEM) serves as a powerful technique for creating shape deformations with the area changes reflecting an underlying density function. In recent decades, DEMs have found widespread applications in fields such as data visualization, geometry processing, and medical imaging. Traditional approaches to the DEM primarily rely on iterative numerical solvers for diffusion equations or optimization-based methods that minimize handcrafted energy functionals. However, these conventional techniques often face several challenges: they may suffer from limited accuracy, produce overlapping artifacts in extreme cases, and require substantial algorithmic redesign when extended from 2D to 3D, due to the derivative-dependent nature of their energy formulations. In this work, we proposed a novel learning-based density-equalizing mapping framework (LDEM) using deep neural networks. Specifically, we introduced a loss function that enforces density uniformity and geometric regularity, and utilized a hierarchical approach to predict the transformations at both the coarse and dense levels. Our method demonstrated superior density-equalizing and bijectivity properties compared to prior methods for a wide range of simple and complex density distributions, and can be easily applied to surface remeshing with different effects. Also, it generalizes seamlessly from 2D to 3D domains without structural changes to the model architecture or loss formulation. Altogether, our work opens up new possibilities for scalable and robust computation of density-equalizing maps for practical applications.
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Yanwen Huang, Lok Ming Lui, Gary P. T. Choi. Learning-based density-equalizing map[J]. AIMS Mathematics, 2025, 10(11): 25756-25790. doi: 10.3934/math.20251140
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